Bias Correction in Localization Problem Yiming (Alex) Ji Research School of Information Sciences and Engineering The Australian National University 1
Collaborators Dr. Changbin (Brad) Yu Professor Brian D. O. Anderson Assistance of Dr. Sam Drake of Australian Defence Science and Technology Organization (DSTO) with original problem formulation and provision of trial data is gratefully acknowledged 2 27 August 2010 CEA Technologies 2
Outline Motivation Bias in Localization Problem Taylor-Jacobian Bias Correction Method Performance Evaluation and Simulation Conclusion 3
Motivation Industry Process control Automation Predictive maintenance Scientific Research High spatial and temporal density sampling Habitat monitoring Event detection Health Care Location aware patient monitoring Patient vital signals Disaster Management Event detection (natural disasters fire, earthquake) Location awareness (fire fighters looking for survivors) Emergency response Military Battlefield surveillance Target tracking 2 April 2009 Thesis Proposal Review 4
Motivation Accurate location of sensors plays a vital role in various network applications 5
Motivation Accurate location of sensors plays a vital role in various network applications Location!!! 6
Motivation Many self-localization algorithms are proposed Generally, localization results are imprecise Environment: noise, non line of sight Hardware: range or angle measuring devices Localization algorithms Many enhancement techniques have been proposed to improve the accuracy of localization Geometric Constraints Error Control Mechanisms (Baoqi Huang) Bias Correction Methods 7
Motivation Many self-localization algorithms are proposed Generally, localization results are imprecise Environment: noise, non line of sight Hardware: range or angle measuring devices Localization algorithms Many enhanced techniques have been proposed to improve the accuracy of localization Geometric Constraints Why we choose bias? Error Control Mechanisms Bias Correction Methods 8
Outline Motivation Bias in Localization Problem Taylor-Jacobian Bias Correction Method Performance Evaluation and Simulation Conclusion 9
What is Bias Bias is a term in estimation theory and is defined as the difference between the expected value of a parameter estimate and the true value of the parameter [1]. [1] J. L. Melsa and D. L. Cohn. Decision and Estimation Theory. McGraw-Hill Inc, 1978. 10
What is Bias Bias is a term in estimation theory and is defined as the difference between the expected value of a parameter estimate and the true value of the parameter [1]. [1] J. L. Melsa and D. L. Cohn. Decision and Estimation Theory. McGraw-Hill Inc, 1978. 11
Bias in Localization Problem In the noisy situation, we assume denotes the localization mapping from the measurements to the target position estimates. We have: where the target location, measurements and measurement noise. denotes the inaccurate estimates of denotes the noisy denotes the 12
Bias in Localization Problem In practice the measurement process will repeated M times. As, we would expect the estimate to go to : Because is nonlinear we have: Therefore the bias appears in the estimation process: 13
Bias in Localization Problem In practice the measurement process will repeated M times. As, we would expect the estimate to go to : The bias will exist if two conditions are satisfied: Because is nonlinear we have: 1. the mapping function is nonlinear 2. the measurements are noisy Therefore the bias appears in the estimation process: 14
Significant Bias in Localization Problem Two sensors at (0, 8) and (0, -8) y value of the target is fixed at 0 while x value changes from 6 to 20 Measurements are bearingonly Different variances used for measurement errors, which are zero mean. 15
Significant Bias in Localization Problem Table 1 and 2 illustrate the bias of the x component compared to the standard deviation of the error in estimating x with different level of noise Bias (before removal) can be a significant fraction of the errors. 16
Significant Bias in Localization Problem Normally the two conditions are satisfied easily: the mapping function is nonlinear the measurements are noisy The bias can be a significant fraction of the error. It is worth to analyse and remove the bias in localization. 17
Outline Motivation Bias in Localization Problem Taylor-Jacobian Bias Correction Method Performance Evaluation and Simulation Conclusion 18
Taylor-Jacobian Bias Correction Method Notations and Assumptions: 1. n denotes the number of dimensions of the ambient space 2. N denotes the number of obtained measurements 3. denotes the position of the target 4. denotes the measurement set 5. denotes measurement errors 6. is the mapping from the target position to the measurements 7. is the localization mapping from the measurements to the target position 19
Formulation of the bias In the noisy case, errors in measurements are inevitable. Therefore the localization problem can be formulated as follows: Next a Taylor series is used to expand the above equation truncating at second order: 20
Formulation of the bias In the noisy The approximate case, errors bias in measurements expression is is immediate: inevitable. Therefore the localization problem can be formulated as follows: Next a Taylor series is used to expand the above equation truncating at second order: 21
Formulation of the bias However it is very difficult to compute the localization mapping and its derivatives. The approximate bias expression is immediate: 22
Formulation of the bias However it is very difficult to compute the localization mapping and its derivatives. The approximate bias expression is immediate: How to analytically express the bias in an easy way? 23
Formulation of the bias However it is very difficult to compute the localization mapping and its derivatives. In contrast can be easily written down! The approximate bias expression is immediate: 24
Different Localization Techniques Range Measurements Bearing-Only Measurements TDOA Measurements 25
Different Localization Techniques Range Measurements How to analytically express the bias in an easy way? Bearing-Only Measurements How to analytically express the bias by TDOA Measurements using f and its derivatives? 26
Taylor-Jacobian Bias Correction Method Jacobian matrix and one of its property are used to calculate the derivatives of g in terms of the derivatives of f. By solving the above equation set, we can obtain the analytical expression for 27
Taylor-Jacobian Bias Correction Method Here we take for example. Assume, differentiating the equation in respect to respectively we can obtain the following equation set: 28
Taylor-Jacobian Bias Correction Method Here we take for example. Assume, differentiating the equation in respect to respectively we can obtain the following equation set: Can be easily expressed analytically 29
Taylor-Jacobian Bias Correction Method Here How we take to analytically for example. express Assume the bias in an, easy differentiating way? the equation in respect to respectively we can obtain the following equation Solved! set: 1. Taylor series 2. Jacobian matrix and its property So we call the proposed method as Taylor-Jacobian bias correction method. 30
Overdetermined Problem Jacobian matrix and one of its property Important assumption: N=n 31
Overdetermined Problem Jacobian matrix and one of its property Important assumption: N=n N>n 32
Overdetermined Problem (N=n+1) Least squares method: 33
Overdetermined Problem (N=n+1) Least squares method: Minimize the distance: 34
Overdetermined Problem (N=n+1) Least squares method: Minimize the distance: For the white point: 35
Overdetermined Problem (N=n+1) Least squares method: Minimize the distance: For the white point: The normal vector: 36
Overdetermined Problem (N=n+1) Least squares method: Finally we can obtain a new mapping: Minimize the distance: For the white point: The normal vector: 37
Overdetermined Problem (N>n+1) With N> n+1, the situation is similar to N=n+1 case except that the extra variable is no longer a scalar. Instead, it is a vector which can be defined as follows: Where denotes a coefficient to minimize the moved distance in each dimension of the normal. 38
Overdetermined Problem (N>n+1) Assume N=4 and n=2. At the white point we can have: These two tangent vectors define a tangent plane 39
Contributions and key mathematic tools of our work Contributions: 1. Express the bias in an easy way by using the function f (mapping from the target position to the measurements) and its derivatives 2. Adopt a method based on least-squares idea to solve the overdetermined problem Mathematic tools: 1. Taylor series 2. Jacobian Matrix 40
Outline Motivation Bias in Localization Problem Taylor-Jacobian Bias Correction Method Performance Evaluation and Simulation Conclusion 41
Simulation Simulation Assumption All simulations are done in two-dimensinol space The three sensors are fixed at (0, 8), (0, -8) and (8,0) The measurement noise for three sensors are produced by i. i. d. Gaussian with zero mean and variance. All the simulation results are obtained from 5000 Monte Carlo experiments. We compare our method with an well-cited bias-correction method GW method [1] [1] M. Gavish and A. J. Weiss. Performance analysis of bearing-only target location algorithms. IEEE Transaction on Aerospace and Electronic Systems, 28(3): 817-827, 1992. 42
Simulation Results - Range Measurement S1 (0,8) d1 Target (x,0) d3 S3 (8,0) d2 S2 (0,-8) Three sensors and a single target Range measurements only Measurement errors are N(0,1) The y value of the target is fixed at 0; x value is adjusted from 6 to 20 43
Simulation Results Bearing-Only Measurement S1 (0,8) Target (x,0) S3 (8,0) S2 (0,-8) Three sensors and a single target Bearing-only measurements Measurement errors are N(0,1) The y value of the target is fixed at 0; x value is adjusted from 6 to 20 44
Simulation Results - Different Level of Noise Truncation of Taylor series is not necessarily justified when the noise is large Noise level is adjusted over a large range via changing the standard deviation of measurement errors, from 0.5 to 3.5 in steps of 0.5 Bearing-only measurements are used S1 (0,8) Target (14,0) S2 (0,-8) 45
Simulation Results - Different Level of Noise Truncation of Taylor series is not necessarily justified when the noise is large Noise level is adjusted over a large range via changing the standard deviation of measurement errors, from 0.5 to 3.5 in steps of 0.5 Bearing-only measurements are used S1 (0,8) Target (14,0) S2 (0,-8) 46
Trial Data Scan-based Measurement S1 (0,5) S3 (0,0) S2 (0,-7) Target (10,0) Three physical sensors and a single target, which is a radar with a mechanically rotating antenna. Two usable sensor measurements are obtained. Noise in measurements is N(0,0.02) 47
Trial Data Scan-based Measurement S1 (0,5) S3 (0,0) S2 (0,-7) Target (10,0) Three physical sensors and a single target, which is a radar with a mechanically rotating antenna. Two usable sensor measurements are obtained. Noise in measurements is N(0,0.02) 48
Performance of the Taylor-Jacobian Method 1. The Taylor-Jacobian method is generic Range Measurement Bearing-only Measurement Scan-based Measurement The performance of the Taylor-Jacobian method is better than the GW method The Taylor-Jacobian method can be more robust to the level of noise than the GW method 49
Outline Motivation Bias in Localization Problem Taylor-Jacobian Bias Correction Method Performance Evaluation and Simulation Conclusion 50
Conclusion Bias arises due to simultaneous presence of noise and nonlinear transformations. In localization, the map need for computing the bias may not be analytically available; its inverse is available so the bias computation needs to be varied A generic Taylor-Jacobian bias correction method is proposed The simulation results demonstrate the performance of the proposed method 51
Thank you! Publications: [1] Y. Ji, C. Yu and B. D. O. Anderson. Bias correction in localization algorithms. IEEE Global Communication Conference, pp. 1-7, 2009. [2] Y. Ji, C. Yu and B. D. O. Anderson. Geometric dilution of localization and bias-correction methods. International Conference on Control & Automation, pp. 578-583, 2010. [3] Y. Ji, C. Yu and B. D. O. Anderson. Localization bias correction in n-dimensional space. IEEE International Conference on Acoustics Speech and Signal Processing, pp. 2854-2857, 2010. [4] Y. Ji, C. Yu and B. D. O. Anderson. Bias-correction method in bearing-only passive localization. European Signal Processing Conference, Published, 2010. [5] Y. Ji, C. Yu and B. D. O. Anderson. Localization bias correction in n-dimensional space. Submitted to IEEE Transaction on Aerospace and Electronic Systems. Contact details: E-mail: yiming.ji@anu.edu.au 52